What Is a Coterminal Angle?

These are angles that have the same initial side, terminal side, and vertex. However, they differ in their numerical measure. This is because one angle may include extra full rotations around the circle.

The angles are different when written as numbers, but they end at the same position when drawn in standard form.

Since one full rotation equals 360° (or 2π radians), coterminal angles are separated by these values.

Example

The angles 30°, 390°, and −330° are coterminal because:

390° = 30° + 360°

−330° = 30° − 360°

All three angles end at the same position on the coordinate plane.

Why Do We Use Coterminal Angles?

These are useful because they make the trigonometric calculations easier and help standardize the angle values. Coterminal angles make it simple to compare results, apply formulas, and interpret graphs. Also, they are widely used in mathematics, physics, and engineering whenever angles repeat after a full rotation.

Coterminal angles are especially useful when:

Working with angles outside the standard range
Comparing angles in trigonometry
Converting between positive and negative angles

How to Find Coterminal Angles

Coterminal angles are found by adding or subtracting full rotations from a given angle.

Coterminal Angles Formula

The following is the formula in degrees and radians.

Coterminal Angles in Degrees

β=α±360∘×k

where k is any integer.

Coterminal Angles in Radians

β=α±2π×k

Example

Find a coterminal angle of 75°.

Positive coterminal angle:

75° + 360° = 435°

Negative coterminal angle:

75° − 360° = −285°

How to Find a Coterminal Angle Between 0 and 360° (or 0 and 2π)?

To find the coterminal angle between 0° and 360°, divide the given angle by 360° and use the remainder.

θstandard=θmod360°

If the result is negative, add 360° to obtain a value in the required range.

For angles measured in radians, use:

θstandard=θmod2π

If the result is negative, add 2.

The final value lies between 0 and 360° (or 0 and 2π) and is coterminal with the original angle.

Example

Find the coterminal angle of 420° between 0° and 360°.

420° ÷ 360° = 1 remainder 60°

So, the required coterminal angle is 60°.

Positive and Negative Coterminal Angles

Coterminal angles can be either positive or negative. They depend on how many full rotations are added or subtracted.

Positive Coterminal Angles

These are obtained by adding full rotations.

Example:

For 140°, positive coterminal angles include:

500°
860°

Negative Coterminal Angles

These are obtained by subtracting full rotations.

Example:

For 140°, negative coterminal angles include:

−220°
−580°

Coterminal Angle Chart

The following are the commonly used angles along with their positive and negative coterminal equivalents.

Coterminal angles of 0°: 360°, 720°, 1080° | −360°, −720°, −1080°

Coterminal angles of 30°: 390°, 750°, 1110° | −330°, −690°, −1050°

Coterminal angles of 45°: 405°, 765°, 1125° | −315°, −675°, −1035°

Coterminal angles of 60°: 420°, 780°, 1140° | −300°, −660°, −1020°

Coterminal angles of 90°: 450°, 810°, 1170° | −270°, −630°, −990°

Coterminal angles of 120°: 480°, 840°, 1200° | −240°, −600°, −960°

Coterminal angles of 135°: 495°, 855°, 1215° | −225°, −585°, −945°

Coterminal angles of 150°: 510°, 870°, 1230° | −210°, −570°, −930°

Coterminal angles of 165°: 525°, 885°, 1245° | −195°, −555°, −915°

Coterminal angles of 180°: 540°, 900°, 1260° | −180°, −540°, −900°

Coterminal angles of 195°: 555°, 915°, 1275° | −165°, −525°, −885°

Coterminal angles of 210°: 570°, 930°, 1290° | −150°, −510°, −870°

Coterminal angles of 225°: 585°, 945°, 1305° | −135°, −495°, −855°

Coterminal angles of 240°: 600°, 960°, 1320° | −120°, −480°, −840°

Coterminal angles of 255°: 615°, 975°, 1335° | −105°, −465°, −825°

Coterminal angles of 270°: 630°, 990°, 1350° | −90°, −450°, −810°

Coterminal angles of 285°: 645°, 1005°, 1365° | −75°, −435°, −795°

Coterminal angles of 300°: 660°, 1020°, 1380° | −60°, −420°, −780°

Coterminal angles of 315°: 675°, 1035°, 1395° | −45°, −405°, −765°

Coterminal angles of 330°: 690°, 1050°, 1410° | −30°, −390°, −750°

Coterminal angles of 345°: 705°, 1065°, 1425° | −15°, −375°, −735°

Coterminal angles of 360°: 720°, 1080°, 1440° | 0°, −360°, −720°

Some Commonly Asked Questions

What is the Coterminal Angle of 1000° between 0° and 360°?

The coterminal angle of 1000° between 0° and 360° is 280°. This is how it is calculated:

1000° ÷ 360° = 2 remainder 280°

So, the coterminal angle is 280°.

How Do I Check If Two Angles Are Coterminal?

Two angles are coterminal if their difference is a multiple of 360°.

Example:

550° − (−170°) = 720°

Since 720° is a multiple of 360°, the angles are coterminal.

Which Angles Are Coterminal with a 125 Degree Angle?

Some coterminal angles of 125° are:

485°
−235°

Coterminal Angle Calculator